{"id":4743,"date":"2021-11-28T15:52:48","date_gmt":"2021-11-28T15:52:48","guid":{"rendered":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/?page_id=4743"},"modified":"2022-01-03T08:11:33","modified_gmt":"2022-01-03T08:11:33","slug":"degree-of-a-polynomial-with-examples-and-faqs","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/degree-of-a-polynomial-with-examples-and-faqs\/","title":{"rendered":"Degree of a Polynomial with Examples and FAQs"},"content":{"rendered":"

[et_pb_section fb_built=”1″ admin_label=”Section” module_class=”mainsec” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#e0f2fd” z_index=”1″ custom_padding=”5px||5px||true|false” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|40px|51px|false|true” custom_padding_tablet=”” custom_padding_phone=”|40px|30px|40px|false|true” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}”][et_pb_column type=”3_5″ admin_label=”Column L” _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_text admin_label=”Acute Angles
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Degree of a Polynomial with Examples and FAQs<\/h1>\n

[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.13.1″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” custom_padding=”15px|15px|54px|4px|false|false” global_colors_info=”{}”]<\/p>\n

What do you mean by Degree of a Polynomial?<\/b><\/h2>\n

The degree of any polynomial is the greatest power of the variable term of that polynomial.<\/span><\/p>\n

For example, look at the polynomial below:<\/span><\/p>\n3 x^{10}+7 x^{5}-12 x^{3}<\/span>\n

Here, x<\/span><\/span> is the variable and the highest power of x<\/span><\/span>\u00a0is 10.<\/span><\/p>\n

Hence, the degree of the polynomial is 10.<\/span><\/p>\n

This was an easy example as the polynomial is in only one variable.<\/span><\/p>\n

Let us take an example of a multivariable polynomial and define its degree.<\/span><\/p>\n

x^{3} y^{4}+y^{6}+x^{2} y^{3}+y^{2}<\/span><\/span><\/p>\n

Here, the first term has a degree equivalent to the sum of exponents of x<\/span> and y<\/span>, i.e.,<\/span><\/p>\n

3 + 4 = 7.<\/span><\/p>\n

The second term has a degree equivalent to the exponent of y<\/span> i.e., 6.<\/span><\/p>\n

The third term again has a degree equivalent to the sum of exponents of x<\/span> and y<\/span>, i.e.,<\/span><\/p>\n

2 + 3 = 5.<\/span><\/p>\n

The last term has a degree equivalent to the exponent of y<\/span>, i.e., 2.<\/span><\/p>\n

Hence the degree of the first term is the highest and therefore the degree of polynomial is 7.<\/span><\/p>\n

<\/span><\/p>\n

Some polynomials with specific types of degrees<\/b><\/h2>\n

<\/b><\/h3>\n

Zero polynomial<\/b><\/h3>\n

The Polynomial consisting of only one term, zero, is known as zero polynomial. The degree of zero polynomial is undefined.<\/span><\/p>\n

For example:<\/strong> 0,0 . x \\text { and } 0 . x^{2}<\/span> <\/span>are some examples of zero polynomial.<\/span><\/p>\n

<\/span><\/p>\n

Let us discuss the various degrees of a polynomial in detail.<\/span><\/p>\n

<\/span><\/p>\n

Constant Polynomial \u2013 Degree \u20180\u2019\u00a0<\/b><\/h3>\n

The polynomial consisting of one term, any constant value without a coefficient, is known as a constant polynomial.<\/span><\/p>\n

For example:<\/strong> 5,23, \\sqrt{29}<\/span> <\/span>are some examples of constant polynomials.<\/span><\/p>\n

The degree of such a polynomial is zero as:<\/span><\/p>\n

5 . x^{0}=5, \\text { as } x^{0}=1<\/span> <\/span>Similarly, it can be shown for the other constants.<\/span><\/p>\n

23 \\cdot y^{0}=23<\/span><\/span><\/p>\n

\\sqrt{29} \\cdot a^{0}=\\sqrt{29}<\/span><\/span><\/p>\n

<\/span><\/p>\n

<\/span><\/p>\n

Linear polynomial<\/b> \u2013 Degree \u20181\u2019<\/b><\/h3>\n

The polynomial with degree 1 is linear polynomial, i.e., the highest exponential power of the variable of this polynomial is 1. The polynomial can have one variable or multiple variables.<\/span><\/p>\n

For example:\u00a0<\/strong><\/p>\n

20 x+5<\/span><\/span> ( Linear polynomial in one variable x<\/span><\/span>)<\/span><\/p>\n

2 x-z<\/span><\/span>\u00a0 \u00a0 (Linear polynomial in two variables x \\text{ and }z<\/span><\/span>)<\/span><\/p>\n

x+y+z<\/span> <\/span>\u00a0(Linear polynomial in three variables<\/span> x,y \\text{ and } z<\/span>)<\/span><\/p>\n

The highest power of any variable in the above examples is \u20181\u2019 only.<\/span><\/p>\n

 <\/p>\n

Quadratic polynomial \u2013 Degree \u20182\u2019<\/b><\/h3>\n

The polynomial with degree 2 is quadratic polynomial, i.e., the highest exponential power of the variable of this polynomial is 2. This type of polynomial can have one variable or multiple variables.<\/span><\/p>\n

For example:<\/strong><\/p>\n

x^{2}-10 x+2<\/span> (Quadratic polynomial in one variable x<\/span><\/span>)<\/span><\/p>\n

xy-y+22<\/span><\/span>\u00a0(Quadratic polynomial in two variables x\\text{ and }y<\/span><\/span>)<\/span><\/p>\n

x y+5 y z-6 z<\/span><\/span>\u00a0(Quadratic polynomial in three variables<\/span> x,y \\text{ and } z<\/span>)<\/span><\/p>\n

The exponent values of two variables in one term of the polynomial are added to find the degree in the last two examples, hence the degree is 2.<\/span><\/p>\n

 <\/p>\n

Cubic polynomial \u2013 Degree \u20183\u2019<\/b><\/h3>\n

The polynomial with degree 3 is a cubic polynomial, i.e., the highest exponential power of the variable of this polynomial is 3. This type of polynomial can again have one variable or multiple variables.<\/span><\/p>\n

For example:<\/strong><\/p>\n

5 x^{3}-14 x+9<\/span> (Cubic polynomial in one variable x<\/span><\/span>)<\/span><\/p>\n

x y^{2}-x^{2} y+12<\/span> (Cubic polynomial in two variables x\\text{ and } y<\/span><\/span>)<\/span><\/p>\n

x y z+12 x^{3}-6 y z<\/span> (Cubic polynomial in three variables<\/span> x,y \\text{ and } z<\/span>)<\/span><\/p>\n

The exponent values of two or three variables in one term of the polynomial are added to find the degree in the last two examples, hence the degree is 3.<\/span><\/p>\n

<\/span><\/p>\n

What is the significance of the degree of a polynomial?<\/b><\/h2>\n

The degree of the polynomial helps to determine the number of zeros or roots of a polynomial function. By looking at the degree we can quickly analyze how many roots of a function exists. The number roots of a polynomial are equal to the degree of the polynomial.<\/span><\/p>\n

For example:<\/strong> f(x)=12 x^{3}-14 x+8<\/span><\/span>, will have three roots for which value of f(x)=0<\/span><\/span>.<\/span><\/p>\n

<\/span><\/p>\n

Example<\/b><\/h2>\n

Identify the degree of the polynomials given below:<\/b><\/p>\n

x^{4}-14 x y^{2}+2 <\/span><\/b><\/p>\n


x^{3}-4 x y^{4}+21<\/span><\/b><\/p>\n

 <\/p>\n

Solution:<\/b><\/p>\n

i. <\/strong>The degree of the polynomial given is 4 as the first term is the one with the highest exponent value and its degree is 4.<\/p>\n

ii. The degree of the second term is the sum of the exponents of x and y, i.e.,<\/span><\/b><\/p>\n

1 + 4 = 5<\/span>, hence the degree of the polynomial is 5.<\/span><\/p>\n

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Explore Other Topics<\/h1>\n

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[\/et_pb_text][et_pb_text admin_label=”Related Concepts
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Related Concepts<\/h1>\n

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\n
Zero Polynomial<\/a><\/div>\n
Quadratic Equation Questions and Solution<\/a><\/div>\n
Linear Equation in one variable<\/a><\/a><\/div>\n<\/div>\n

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Frequently Asked Questions\u00a0<\/span><\/span><\/h1>\n
    <\/ol>\n

    1. What do you mean by the degree of a polynomial?
    <\/strong><\/h3>\n

    Ans: <\/strong>The degree of any polynomial is the greatest power of the variable term of that polynomial.
    <\/strong><\/span><\/p>\n

    For example:\u00a0<\/span><\/p>\n

    4 x^{4}+7 x^{5}-22 x^{3}<\/span><\/span><\/p>\n

    Here, x<\/span><\/span> is the variable and the highest power of x<\/span><\/span> is 5, hence the degree of the polynomial is \u20185\u2019.<\/span><\/span><\/p>\n

    2. What is the degree of a constant polynomial?<\/strong><\/h3>\n

    Ans: <\/strong>The degree of a constant polynomial is zero. As there is no variable term present the power of the variable is considered to be zero.<\/span><\/span><\/b><\/p>\n

    3. Is the degree of a zero polynomial defined?<\/b><\/h3>\n

    Ans:\u00a0 <\/b>No degree of a zero polynomial cannot be defined, i.e., it is undefined.<\/span><\/p>\n

    [\/et_pb_text][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"

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